## Abstract

A general method of surface profiling with phase-shifting interferometry techniques uses iterative linear regression to fit the sequence of interferograms to a physical model of the cavity. The physical model incorporates all important cavity influences, including environmentally induced rigid-body motion, phase shifter miscalibrations, multiple interference, geometry-induced spatial phase-shift variations, and their cross-couplings. By incorporating an initial estimate of the surface profile and iteratively solving for space- and time-dependent variables separately, convergence is robust and rapid. The technique has no restriction on surface shape or departure.

© 2014 Optical Society of America

Full Article |

PDF Article
### Equations (14)

Equations on this page are rendered with MathJax. Learn more.

(1)
$$I(\mathbf{x},t)={\left|\frac{\sqrt{{r}_{\text{ref}}}+\sqrt{{r}_{\text{test}}}\text{\hspace{0.17em}}\mathrm{exp}[i\mathrm{\Theta}(\mathbf{x},t)]}{1+\sqrt{{r}_{\text{ref}}{r}_{\text{test}}}\text{\hspace{0.17em}}\mathrm{exp}[i\mathrm{\Theta}(\mathbf{x},t)]}\right|}^{2},$$
(2)
$$I(\mathbf{x},t)=A(\mathbf{x})+V(\mathbf{x})\sum _{k=1}^{K}{(-g)}^{k-1}\text{\hspace{0.17em}}\mathrm{cos}[k\mathrm{\Theta}(\mathbf{x},t)],$$
(3)
$$\mathrm{\Theta}(\mathbf{x},t)=\mathrm{\Phi}(\mathbf{x})+\mathrm{\Delta}(\mathbf{x},t),$$
(4)
$$\mathrm{\Delta}(\mathbf{x},t)=\phi ({t}_{0})+\alpha (t)x+\beta (t)y+[\phi (t)-\phi ({t}_{0})]\sqrt{1-\rho {(\mathbf{x})}^{2}{c}^{2}},$$
(5)
$$A(\mathbf{x})=\{\mathrm{max}[{I}_{m}(\mathbf{x},t)]+\mathrm{min}[{I}_{m}(\mathbf{x},t)]\}/2\phantom{\rule{0ex}{0ex}}V(\mathbf{x})=\{\mathrm{max}[{I}_{m}(\mathbf{x},t)]-\mathrm{min}[{I}_{m}(\mathbf{x},t)]\}/2,$$
(6)
$${\phi}_{n}(t)=\phi (t)+{\phi}^{\prime}(t)\phantom{\rule{0ex}{0ex}}{\alpha}_{n}(t)=\alpha (t)+{\alpha}^{\prime}(t)\phantom{\rule{0ex}{0ex}}{\beta}_{n}(t)=\beta (t)+{\beta}^{\prime}(t)\phantom{\rule{0ex}{0ex}}{c}_{n}=c+{c}^{\prime},$$
(7)
$${I}_{n}(\mathbf{x},t)=I(\mathbf{x},t)+\cdots +[{\phi}^{\prime}(t)\gamma (\mathbf{x})+{\alpha}^{\prime}(t)x+{\beta}^{\prime}(t)y+{c}^{\prime}\eta (\mathbf{x},t)]H(\mathbf{x},t),$$
(8)
$$H(\mathbf{x},t)=-V(\mathbf{x})\sum _{k=1}^{K}{(-g)}^{k-1}k\text{\hspace{0.17em}}\mathrm{sin}[k\mathrm{\Theta}(\mathbf{x},t)],$$
(9)
$$\gamma (\mathbf{x})=\sqrt{1-\rho {(\mathbf{x})}^{2}{c}^{2}}\phantom{\rule{0ex}{0ex}}\eta (\mathbf{x},t)=-[\phi (t)-\phi ({t}_{0})]\rho {(\mathbf{x})}^{2}c/\gamma (\mathbf{x}).$$
(10)
$$\chi (t)=\frac{1}{2M}\sum _{i=1}^{M}{\left[\frac{{I}_{m}({\mathbf{x}}_{i},t)-{I}_{n}({\mathbf{x}}_{i},t)}{V({\mathbf{x}}_{i})}\right]}^{2},$$
(11)
$$I(\mathbf{x},t)=A(\mathbf{x})+\sum _{k=1}^{K}{C}_{k}(\mathbf{x})\mathrm{cos}[k\mathrm{\Delta}(\mathbf{x},t)]+\sum _{k=1}^{K}{S}_{k}(\mathbf{x})\mathrm{sin}[k\mathrm{\Delta}(\mathbf{x},t)],$$
(12)
$${C}_{k}(\mathbf{x})=V(\mathbf{x}){(-g)}^{k-1}\text{\hspace{0.17em}}\mathrm{cos}[k\mathrm{\Phi}(\mathbf{x})]\phantom{\rule{0ex}{0ex}}{S}_{k}(\mathbf{x})=-V(\mathbf{x}){(-g)}^{k-1}\text{\hspace{0.17em}}\mathrm{sin}[k\mathrm{\Phi}(\mathbf{x})].$$
(13)
$$V(\mathbf{x})=\sqrt{{C}_{1}{(\mathbf{x})}^{2}+{S}_{1}{(\mathbf{x})}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{\Phi}(\mathbf{x})={\mathrm{tan}}^{-1}[-{S}_{1}(\mathbf{x})/{C}_{1}(\mathbf{x})],$$
(14)
$$g=\frac{\sum _{i=1}^{M}\sqrt{{C}_{2}{({\mathbf{x}}_{i})}^{2}+{S}_{2}{({\mathbf{x}}_{i})}^{2}}}{\sum _{i=1}^{M}V({\mathbf{x}}_{i})}.$$